The first gap for minimal submanifolds in spheres
Abstract
Let f:Mn→Snn+q(1) be a minimal immersion of a closed connected manifold with n≥3 and q≥2, and let S be the squared length of its second fundamental form. We prove an integral inequality that gives maxM S≥ n if the immersion is not totally geodesic. If n≠4 or q=2, equality maxM S=n holds if and only if the immersion is a Clifford minimal hypersurface; if n=4 and q≥3, equality holds if and only if the immersion is a Clifford minimal hypersurface or the standard minimal embedding of ℂℙ2 in Sn7(1). Both classifications are understood up to an ambient isometry, a totally geodesic inclusion, and a finite covering. We construct the integral inequalities using polynomial weights and divergence identities. The required algebraic estimates are proved by sums of squared tensor norms.
Related
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BibTeX
@misc{ge2026first,
title = {{The first gap for minimal submanifolds in spheres}},
author = {Jianquan Ge and Kai Jia and Fagui Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08126},
eprint = {2611.08126},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08126},
}
RIS
TY - UNPB
TI - The first gap for minimal submanifolds in spheres
AU - Jianquan Ge
AU - Kai Jia
AU - Fagui Li
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08126
UR - https://arxiv.org/abs/2611.08126
AB - Let f:M^n→Sn^n+q(1) be a minimal immersion of a closed connected manifold with n≥3 and q≥2, and let S be the squared length of its second fundamental form. We prove an integral inequality that gives max_M S≥ n if the immersion is not totally geodesic. If n≠4 or q=2, equality max_M S=n holds if and only if the immersion is a Clifford minimal hypersurface; if n=4 and q≥3, equality holds if and only if the immersion is a Clifford minimal hypersurface or the standard minimal embedding of ℂℙ^2 in Sn^7(1). Both classifications are understood up to an ambient isometry, a totally geodesic inclusion, and a finite covering. We construct the integral inequalities using polynomial weights and divergence identities. The required algebraic estimates are proved by sums of squared tensor norms.
ER -